Mass transfer term

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

Heat-Transfer Applications Heat-transfer analogs of common mass-transfer terms are ... 

When u E, this interstitial mixing effect was considered complete, and the resistance to mass transfer in the mobile phase between the particles becomes very small and the equation again reduces to the Van Deemter equation. However, under these circumstances, the C term in the Van Deemter equation now only describes the resistance to mass transfer in the mobile phase contained in the pores of the particles and, thus, would constitute an additional resistance to mass transfer in the stationary (static mobile) phase. It will be shown later that there is experimental evidence to support this. It is possible, and likely, that this was the rationale that explains why Van Deemter et al. did not include a resistance to mass transfer term for the mobile phase in their original form of the equation. 

It is now seen that only the resistance to the mass transfer term for the stationary phase is position dependent. All the other terms can be used as developed by Van Deemter, providing the diffusivities are measured at the outlet pressure (atmospheric) and the velocity is that measured at the column exit. 

The resistance to the mass transfer term for the stationary phase will now be considered in isolation. The experimentally observed plate height (variance per unit length) resulting from a particular dispersion process [e.g., (hs), the resistance to... 

It is a common procedure to assume certain conditions for the chromatographic system and operating conditions and, as a result, simplify equations (20) and (21). However, in many cases the assumptions can easily be over-optimistic, to say the least. It is necessary, therefore, to carefully consider the conditions that may allow such simplifying procedures and take steps to ensure that such conditions are carefully met when such expressions are used in practice. Now, the relative magnitudes of the resistance to mass transfer terms will vary with the type of columns (packed or capillary), the type of chromatography (GC or LC) and the type of particle, i.e., porous or microporous (diatomaceous support or silica gel). 

It is seen that the Van Deemter equation predicts that the total resistance to mass transfer term must also be linearly related to the reciprocal of the solute diffusivity, either in the mobile phase or the stationary phase. Furthermore, it is seen that if the value of (C) is plotted against 1/Dni, the result will be a straight line and if there is a... 

Figure 7. Graph of Resistance to Mass Transfer Term against the Reciprocal of the Solute Diffusivity in the Mobile Phase...

In Figure 7, the resistance to mass transfer term (the (C) term from the Van Deemter curve fit) is plotted against the reciprocal of the diffusivity for both solutes. It is seen that the expected linear curves are realized and there is a small, but significant, intercept for both solutes. This shows that there is a small but, nevertheless, significant contribution from the resistance to mass transfer in the stationary phase for these two particular solvent/stationary phase/solute systems. Overall, however, all the results in Figures 5, 6 and 7 support the Van Deemter equation extremely well. 

The predicted linear relationship between the resistance to mass transfer term and the square of the particle diameter is clearly demonstrated in Figure 8. The linear... 

Now, at high linear velocities, the longitudinal diffusion term will become insignificant and, equally important, the resistance to mass transfer term that incorporates the inverse function of diffusivity will become large, thus improving the precision of measurement. 

Now, it is of interest to determine if either the resistance to mass transfer term for the mobile phase or, the resistance to mass transfer term in the stationary phase dominate in the equation for the variance per unit length of a GC packed column. Consequently, taking the ratio of the two resistance to mass transfer terms (G)... 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

Thus, the resistance to mass transfer term for the static mobile phase will be... 

It is seen that the peak capacity realized is far less than would be expected from the approximate calculation. This, in fact, is not surprising due to the size of the solute molecules. The diffusivity of the large solute molecules in either phase is so low that the resistance to mass transfer terms become inordinately large. Consequently, when operating significantly above the optimum velocity, very poor efficiencies are obtained. 

Note that the accumulation and reaction terms are based on the volume of the liquid phase but that the mass transfer term is based on the working volume, V=Vi + Vg. The gas-phase balance is... 

The mass transfer term in this equation is indeterminate since ki 00 and a — Ug 0. The indeterminacy is overcome by using Equation (11.5). Thus,... 

The central difficulty in applying Equations (11.42) and (11.43) is the usual one of estimating parameters. Order-of-magnitude values for the liquid holdup and kiA are given for packed beds in Table 11.3. Empirical correlations are unusually difficult for trickle beds. Vaporization of the liquid phase is common. From a formal viewpoint, this effect can be accounted for through the mass transfer term in Equation (11.42) and (11.43). In practice, results are specific to a particular chemical system and operating mode. Most models are proprietary. 

The first of these factors pertains to the complications introduced in the rate equation. Since more than one phase is involved, the movement of material from phase to phase must be considered in the rate equation. Thus the rate expression, in general, will incorporate mass transfer terms in addition to the usual chemical kinetics terms. These mass transfer terms are different in type and number in different kinds of heterogeneous systems. This implies that no single rate expression has a general applicability. 

Experimental values for several systems are given by Cornell et al. (1960), Eckert (1963), and Vital et al. (1984). A selection of values for a range of systems is given in Table 11.3. The composite mass transfer term KGa is normally used when reporting experimental mass-transfer coefficients for packing, as the effective interfacial area for mass transfer will be less than the actual surface area a of the packing. 

Inclusion of this mass transfer term results in the propagation equation... 

While the mass balances given above are relatively straightforward (assuming that a suitable closure can be derived for the mass-transfer terms), the momentum balances are significantly more complicated. In their simplest forms, they can be written as follows ... 

As mentioned earlier, since the interfaces between phases are not resolved in the CFD model, the Reynolds-average mass-transfer terms and the... 

Three special cases of equation 9.2-18 arise, depending on the relative magnitudes of the two mass-transfer terms in comparison with each other and with the reaction term (which is always present for reaction in bulk liquid only). In the extreme, if all mass-transfer resistance is negligible, the situation is the same as that for a homogeneous liquid-phase reaction, ( rA)im = kAcAcB. 

Ctl is the mass transfer term and arises because of the finite time taken for solute molecules to move between the two phases. Consequently, a true equilibrium situation is never established as the solute moves through the system, and spreading of the concentration profiles results. The effect is minimal for small particle size and thin coatings of stationary phase but increases with flow rate and length of column or surface. 

The particle size of a solid support is critical in striking a compromise between column efficiency and speed of separation. Both the multiple path term A and the mass transfer term (CSl of equation (4.46) (p. 89)) are reduced by reducing particle size thus leading to increased efficiency. However, as particle size is reduced, the pressure drop across the column must be increased if a reasonable flow rate is to be maintained. The optimum particle sizes for 1/8 in columns are 80/100 or 100/120 mesh and for 1/4 in columns 40/60 or 60/80 mesh. 

B = Longitudinal molecular diffusion in both mobile and stationary phases, and C = Kinetic or mass transfer term originating in the stationary phase. 

Laubriet et al. [Ill] modelled the final stage of poly condensation by using the set of reactions and kinetic parameters published by Ravindranath and Mashelkar [112], They used a mass-transfer term in the material balances for EG, water and DEG adapted from film theory J = 0MMg — c ), with c being the interfacial equilibrium concentration of the volatile species i. 

B, molecular diffusion term C, resistance to mass transfer term D summation for gas chromatography and E, summation for liquid chromatography. 

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